User talk:Ikosarakt1/Optimized notation
Great. Except, in Rule 11, where does M come from? King2218 (talk) 14:35, May 12, 2014 (UTC) :Example could helps: A(B(1),B(1)+B(1)+B(1))1+1+1 = A(B(1),B(1)+B(1)+B(1)1+1+1) = A(B(1),M+B(1)1+1+1) = A(B(1),M+A(B(1),M+B(1)1+1)). Here we take: M = B(1)+B(1) and transform B(1)1+1+1 to A(B(1),M+B(1)1+1). Ikosarakt1 (talk ^ ) 19:31, May 12, 2014 (UTC) You should mention it on your user page in that case. There is no M on the left hand side, and it appears out of blue on the right hand side. LittlePeng9 (talk) 19:35, May 12, 2014 (UTC) :Well, my goal is remove English from rules, so all things must be intuitively understandable. That makes it a bit ill-defined, so my notation is under construction for now. Ikosarakt1 (talk ^ ) 20:13, May 12, 2014 (UTC) ::I think A(B(L),M+B(N)n+1) = A(B(L),M+A(B(N),M+B(N))n) should work. I guess. King2218 (talk) 07:01, May 13, 2014 (UTC) ::No, that won't work. A(B(0),A(B(1),B(0)1+1)) is supposed to be A(B(0),A(B(1),A(B(0),A(B(1),B(0)1))), but your rule says that it is just A(B(0),A(B(1),A(B(0)1))). It is like we admit that \psi(\psi_{\Omega_2}(\Omega)) = \psi(\psi_{\Omega_2}(\psi(\psi(\psi(\cdots)))) instead of \psi(\psi_{\Omega_2}(\psi(\psi_{\Omega_2}(\psi(\psi_{\Omega_2}(\cdots))))) . Ikosarakt1 (talk ^ ) 19:26, May 13, 2014 (UTC) ::Oops. King2218 (talk) 07:29, May 14, 2014 (UTC) This notation is ill-defined because you don't give any clue on how to get this M in rule 11. It's not "a bit" ill-defined, it's almost completely ill-defined past \(\varepsilon_0\). LittlePeng9 (talk) 20:04, May 13, 2014 (UTC) :Probably the rule A(B(N),M+B(N)n+1) = A(B(N),M+A(B(N),M+B(N)n)) (N = 0 | N = K+1) makes more sense. Ikosarakt1 (talk ^ ) 20:10, May 13, 2014 (UTC) I wonder what happens if you put a C such as: (A(B(C(2),2),2),2). King2218 (talk) 07:37, May 14, 2014 (UTC) :I shall try to manage C(N) so that it will work as N-th Mahlo. B-function will be ternary and works like \chi function. The first argument will be in the form C(N) (indicating that C(N) is the diagonalizer of that function), second and third in the form of normal structures like in \chi(\alpha,\beta) . Your structure must be written as A(B(C(1+1),C(1+1),1+1),1+1) and it could be comparable to \psi(\chi(\chi_{M_2}(M_2,2))) . ::Hold on, how does \chi_{M_2} work? The cardinals that it can generate don't seem to be obvious. King2218 (talk) 14:59, May 14, 2014 (UTC) ::Note how we defined \theta_{\Omega_2} : \theta_{\Omega_2}(0,\alpha) = \varepsilon_{\Omega+1+\alpha} . In \chi function, we can let \chi_{M_2}(0,\alpha) = \Omega_{M+1+\alpha} . Generally, \chi_{M_{\alpha+1}}(0,\beta) = \Omega_{M_\alpha+1+\beta} . Now I'm looking how to transform it to my notation. Ikosarakt1 (talk ^ ) 15:30, May 14, 2014 (UTC) ::Well that's great! King2218 (talk) 16:45, May 14, 2014 (UTC) Further, C(N) will be extented to C(D(N),M,L,K), and D(N) must be comparable to N-th compact cardinal so it will marks the limit of Deedlit's notation. Final goal of this notation, when E(N), F(N), ... (arbitrary letter-level) ... will be defined, is overgrowing Loader's function. But that's only in idea, the rules may be complicated, so I can't write it down for now. Ikosarakt1 (talk ^ ) 09:41, May 14, 2014 (UTC) : I don't want to crash your dreams, but it's really doubtful you will get to Loader's function level. Many mathematicians tried, no one has got even close to full power of it. LittlePeng9 (talk) 13:12, May 14, 2014 (UTC) :My main problem is that I can't make an isomophism between Loader's structures and mine. So even if I'd make 64-rule notation or even 256-rule notation, I can't be sure that I defined the function which grows faster than D(n) because I have no clue where it goes in terms of CoC. :Also, maybe mathematicians are just tired by more practical works than just trying to surpass "the fastest computable function ever defined"? Anyways, the fact that we can reach the level D(n) just having enough perseverance for defining more are more rules and diagonalizers, seems encouraging. Ikosarakt1 (talk ^ ) 13:23, May 14, 2014 (UTC) Minor errors: Rule 13 must be B(L,N+1)n = B(Ln,B(L,N)) and Rule 14 must be B(L,0)nn,0). Also, what will happen if we encounter, say, B(1,0)n or A(B(M,N),0) for all N not in the form of @+1? King2218 (talk) 15:12, May 14, 2014 (UTC) :Thanks for noting, I removed all these problems. My notation now has 16 rules. Ikosarakt1 (talk ^ ) 16:43, May 14, 2014 (UTC) :Oh, and you should have reached \chi function with Mahlo because you wrote that the limit ordinal is \psi(\chi(M,0)) which evaluates to \psi(\psi_{\chi(M,0)}(\psi_{\chi(M,0)}(\psi_{\chi(M,0)}(...)))) or perhaps you meant that it had limit ordinal \psi(\psi_{\chi(M,0)}(0)) . King2218 (talk) 16:59, May 14, 2014 (UTC) : \psi(\chi(M,0))n = \psi(\chi(\chi(\chi(\cdots))),0) (n \chi 's), approaching the limit of my notation. By the way, I don't think that \psi_{\chi(M,0)}(\alpha) is sensible because M diagonalizes through \chi function, not \psi one. Ikosarakt1 (talk ^ ) 17:10, May 14, 2014 (UTC) :"Nope. I think that \chi(M,0) is the diagonalizer of the \psi_{\chi(M,0)} function. (and therefore \psi_{\chi(M)}(0)=\chi(\chi(\chi(...))) ) It's like how I(2) is bigger than I_I_I_I_..." <--- I could have said that but I just noticed that M is the diagonalizer of the \chi function. But still, the \chi function is quite different from the \psi function. :Actually, in Deedlit's blog post, he gives an example with \psi_{I(1,0)} which is equal to \psi_{\chi(M)} . King2218 (talk) 17:24, May 14, 2014 (UTC) :I have doubts that \chi(\alpha,0) for limit \alpha and \chi(\alpha,\beta) for limit \beta are even diagonalizers and corresponding \psi_{\chi(\alpha,\beta)}(\lambda) exists for them. They look as limits of diagonalizers instead. Take relatively small example: \Omega_\omega . How \psi_{\Omega_\omega}(\alpha) must work then? Ikosarakt1 (talk ^ ) 19:03, May 14, 2014 (UTC) :For that I use either \psi_{\omega} or \psi_{\Omega_{\omega+1}} . Then, \psi_{\Omega_{\omega+1}}(\alpha) is the (1+\alpha) th fixed point of the function f:\alpha \mapsto \Omega_{\omega}^{\alpha} . King2218 (talk) 09:45, May 16, 2014 (UTC) :I think that only regular cardinals can be diagonalizers. King2218 (talk) 16:53, May 18, 2014 (UTC) :Oops, never mind. I didn't read Deedlit's blog post carefully. Sorry. :P King2218 (talk) 05:13, May 19, 2014 (UTC)